# Blog Archives

# Topic Archive: admissible sets

# Set-theoretic mereology as a foundation of mathematics

In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, it is natural to wonder whether one might find a similar success for set-theoretic mereology, based upon the set-theoretic inclusion relation $subseteq$ rather than the element-of relation $in$. How well does set-theoretic mereological serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics in terms of the subset relation to the same extent that set theorists have argued (with whatever degree of success) that we may find faithful representations in terms of the membership relation? Basically, can we get by with merely $subseteq$ in place of $in$? Ultimately, I shall identify grounds supporting generally negative answers to these questions, concluding that set-theoretic mereology by itself cannot serve adequately as a foundational theory.

This is joint work with Makoto Kikuchi, and the talk is based on our joint article:

J. D. Hamkins and M. Kikuchi, Set-theoretic mereology, Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, pp. 1-24, 2016.

# Admissible Covers and Compactness Arguments for Ill-founded Models of Set Theory

The Barwise compactness theorem is a powerful tool, allowing one to prove many interesting results which cannot be gotten just from the ordinary compactness theorem. For example, it can be used to show that every countable transitive model of set theory has an end extension which is a model of $V = L$. However, the Barwise compactness theorem only applies to transitive sets. What are we to do if we want to have compactness arguments for ill-founded models of set theory?

This is where Barwise’s notion of the admissible cover of a (possibly ill-founded) model of set theory comes in. In this talk, we will see how to construct admissible covers and how they can be used to extend compactness arguments to the ill-founded case.

This talk is a prequel of sorts to my next talk in this seminar. Some of the results discussed in this talk will play a crucial roll in the arguments there.